On the derivation of boundary integral equations for ... - Inside Mines

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 39, NUMBER 2

FEBRUARY 1998

On the derivation of boundary integral equations for scattering by an infinite two-dimensional rough surface J. A. DeSanto Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401-1887

P. A. Martin Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

~Received 9 June 1997; accepted for publication 17 September 1997! A plane acoustic wave is incident upon an infinite, rough, impenetrable surface S. The aim is to find the scattered field by deriving a boundary integral equation over S, using Green’s theorem and the free-space Green’s function. This requires careful consideration of certain integrals over a large hemisphere of radius r; it is known that these integrals vanish as r→` if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane waves must be present. It is shown that the well-known Helmholtz integral equation is not valid in all circumstances. For example, it is not valid when the scattered field includes plane waves propagating away from S along the axis of the hemisphere. In particular, it is not valid for the simplest possible problem of a plane wave at normal incidence to an infinite flat plane. Some suggestions for modified integral equations are discussed. © 1998 American Institute of Physics. @S0022-2488~98!01302-4#

I. INTRODUCTION

A bounded three-dimensional obstacle with a smooth surface S is surrounded by a compressible fluid. A plane time-harmonic sound wave is incident on the obstacle; the problem is to calculate the scattered field u. In order to have a well-posed boundary-value problem ~with existence and uniqueness!, one imposes the Sommerfeld radiation condition,

r

S

D

]u 2iku →0 as r→`, ]r

~1!

uniformly in all directions. Here r is a spherical polar coordinate, k is the wave number, and the time-dependence is e 2i v t . Physically, the radiation condition ensures that the scattered waves propagate outwards, away from the obstacle. A well-known method for solving the above problem is to derive a boundary integral equation for the boundary values of u on S. In the derivation, Green’s theorem is applied to u and a fundamental solution G, in the region bounded internally by S and externally by C r , a large sphere of radius r. It turns out that the radiation condition implies that the integral

I ~ u;C r ! [

ES Cr

u

D

]G ]u 2G ds→0 as r→`, ]r ]r

~2!

and so only boundary integrals over S remain. For more information, see Colton and Kress.1 Assume that S is a sound-hard surface ~Neumann condition!. Then, the method described above leads to the following boundary integral equation:

u~ p !2

E

u~ q !

S

0022-2488/98/39(2)/894/19/$15.00

]G ~ p,q ! ds q 5 ]nq 894

E ]]

u inc

S

nq

G ~ p,q ! ds q , pPS,

~3!

© 1998 American Institute of Physics

Copyright ©2001. All Rights Reserved.

J. Math. Phys., Vol. 39, No. 2, February 1998

J. A. DeSanto and P. A. Martin

895

here, u inc is the given incident wave. One can also derive an equation for the boundary values of the total field u tot5u inc1u; this boundary integral equation is u tot~ p ! 2

E

S

u tot~ q !

]G ~ p,q ! ds q 52u inc~ p ! , pPS. ]nq

~4!

We shall refer to Eqs. ~3! and ~4! as standard Helmholtz integral equations. Similar equations can be derived for sound-soft surfaces (u tot50 on S). Suppose now that the obstacle is unbounded. The prototypical problem is scattering ~reflection! of a plane wave by an infinite flat plane, S. As is well known, the incident wave is reflected specularly as a single propagating plane wave. More generally, if S is an infinite rough surface, an incident plane wave will be scattered into a spectrum of plane waves. For such problems, the Sommerfeld radiation condition is definitely inappropriate as it is not satisfied by a plane wave. Nevertheless, it is customary to proceed, assuming that the scattered field can be represented in terms of plane waves, at least at some distance from S. Typically, this requires the discarding of an integral such as ~2!, but with the large sphere C r replaced by a large hemisphere H r . This paper began as an attempt to justify this step. In a previous paper,2 we derived boundary integral equations of Helmholtz type for onedimensional rough surfaces. We found that the standard Helmholtz integral equations are valid, except that the right-hand side of Eq. ~4! must be replaced by u inc( p) for grazing incidence. It is perhaps surprising that an analysis of this kind has not been given before: most authors have been content to write down an integral equation such as Eq. ~4!, prior to extensive numerical computations. However, it turns out that the necessary analysis for scattering by a twodimensional rough surface is not straightforward and, moreover, it yields some surprises. For example, the simplest problem, namely reflection of a plane wave at normal incidence upon a flat surface, leads to divergent integrals: the standard Helmholtz integral equation ~3! is not valid for this problem. The paper is organized as follows. Section II is devoted to formulating the problem, with some background on angular-spectrum representations and integral representations ~using G). Green’s theorem is applied inside a volume whose closed boundary is made up of three pieces: the large hemisphere H r ; a large circular piece, S r , of the rough surface; and a cylindrical surface T r , joining S r and H r . Estimation of integrals over H r is carried out in Secs. III–V. Thus the method of stationary phase for multiple integrals and an expansion method are used in Secs. III and IV, respectively, but only for a single plane wave. Results for I(u;H r ) are obtained in Sec. V. The contribution from integrating over T r is considered in Sec. VI. Unlike in two dimensions ~onedimensional rough surface!, this contribution may not be negligible; it is evaluated under additional, but reasonable, a priori assumptions on the form of the scattered field near S. This is a weakness of the present analysis. Finally, boundary integral equations of the Helmholtz type are derived in Sec. VII. Further work is needed to tighten up the analysis and to investigate the numerical consequences.

II. FORMULATION

Consider the scattering of a plane wave by a two-dimensional rough surface, S, described by z5s ~ x,y ! , 2`,x,`, 2`,y,` with 2h,s(x,y)0. The acoustic medium occupies z.s and, for definiteness, we assume that S is a smooth, sound-hard surface. Thus we can write the total field as u tot5u inc1u, where u is the scattered field. The incident plane wave is u inc~ r, u , f ! 5exp$ iki •x% , 0< u i < 21 p ,


~5!

896



where ki 5k(sinui ,0,2cosui), u i is the angle of incidence ~it is the angle between the direction of propagation and the negative z-axis!, x5rxˆ5r ~ sinu cosf ,sinu sinf ,cosu ! , and (r, u , f ) are spherical polar coordinates: x5r sinu cosf, y5r sinu sinf and z5r cosu. All the fields u tot , u inc , and u satisfy the Helmholtz equation, ~ ¹ 2 1k 2 ! u50

for z.s. The boundary condition is

] u tot ]n

50

on S,

~6!

where ] / ] n denotes normal differentiation out of the acoustic medium. A. Reflection by a flat surface

It is instructive to consider the very simple problem of reflection by a flat surface, so that s50. The textbook solution for the scattered field is u ~ r, u , f ! 5exp$ iks•x% for 0< u i , 21 p ,

~7!

where ks5k(sinui ,0,cosui). When u i 5 21 p ~‘‘grazing incidence’’!, we have u[0: the incident wave satisfies the boundary condition on S. So, for 0< u i , 21 p , u tot52 e ikxsinu i cos~ kzcosu i ! solves the problem. But consider u 8tot[u tot1u g

~8!

with u g 5V ~ b ! e ik ~ xcosb 1ysinb ! , where b and V( b ) are arbitrary, with 2 p , b < p . u 8tot also ‘‘solves’’ the problem, in that it satisfies the Helmholtz equation and the boundary condition. Of course, we disallow this second solution, unless V[0: but why? The answer is because of the radiation condition ~which we have yet to specify!. For example, take b 50 and V(0)51, so that u g 5e ikx ; this gives an ‘‘outgoing’’ grazing wave at x51` but it is an ‘‘incoming’’ grazing wave at x52`, we must therefore exclude it. Indeed, we must exclude all contributions u g , for any b and V. A similar condition is imposed on the two-dimensional problem.3 However, the threedimensional problem has another feature, for we could consider replacing u g in Eq. ~8! by 1 2p

E

p

V ~ b ! e ik ~ xcosb 1ysinb ! d b ,

2p

where V is a continuous function; but, as u g has been excluded, we must also exclude all linear combinations of such plane grazing waves. In particular, by taking V( b )5(2i) n e in b , we see that we must exclude the cylindrical standing waves J n ~ kR ! e in f ,

~9!

where J n is a Bessel function,4 R5r sinu, and (R, f ,z) are cylindrical polar coordinates of the point at x. On the other hand, the exact scattered field, given by Eq. ~7!, when evaluated on any plane z5constant, has an azimuthal Fourier component proportional to




J n ~ k i R ! e in f ,

897

~10!

where k i 5k sinui,k. Thus if one wants to formulate a radiation condition, mathematically, it must be such that fields ~9! are excluded but fields ~10! are permitted. This discussion suggests that the specification of a mathematical radiation condition for the present class of problems ~plane-wave scattering by an infinite two-dimensional rough surface! will not be straightforward. However, the physical purpose of a radiation condition is clear: it is to exclude all ‘‘incoming’’ waves apart from the incident wave. We shall return to radiation conditions in Sec. II B.

B. Angular-spectrum representations

For any rough surface S, the scattered field in the half-space z.0 may be written using an angular-spectrum representation, u ~ x,y,z ! 5 5

E E E E `

`

2`

2`

p /2

p

0

F ~ m , n ! e ik ~ m x1 n y1mz !

2p

dm dn m~ k !

A ~ a , b !v~ r, u , f ; a , b ! d a d b 1evanescent terms.

~11!

Here F is the spectral amplitude, A( a , b )5F(sina cosb, sina sinb), k 5 Am 2 1 n 2 , and m~ k !5

H

A12 k 2 , 0< k ,1, i Ak 2 21, k .1,

the function v is defined by 1 v~ r, u , f ; a , b ! 5exp$ ik•x% , 0< a < p , u b u < p , 2

~12!

where k5kkˆ5k ~ sina cosb ,sina sinb ,cosa ! . The integrals are superpositions of plane waves; they are propagating, homogeneous plane waves when 0< k ,1, and they are evanescent, inhomogeneous plane waves when k .1. In Eq. ~11!, we see the propagating plane waves explicitly: they propagate in the direction of kˆ, with an ~unknown! complex amplitude, A( a , b ); the ‘‘evanescent terms’’ decay exponentially with z. For more information on angular-spectrum representations, see Clemmow5 and DeSanto and Martin.6 In general, the spectral amplitude must be considered as a generalized function. Thus it is convenient to extract a continuous component from F, writing the scattered field as ~13!

u5u pr1u ev1u con , where N

u pr~ r, u , f ! 5

(

n50

A n v~ r, u , f ; a n , b n ! ,

M

u ev~ r, u , f ! 5

u con~ x,y,z ! 5

E E

(

m51

`

`

2`

2`

B m w ~ r, u , f ; m m , n m ! ,

C ~ m , n ! e ik ~ m x1 n y1mz !

dm dn , m~ k !


~14!

~15!

898



w ~ r, u , f ; m , n ! 5exp$ ikr sinu @ m cosf 1 n sinf # 2kr cosu

Ak 2 21 % ,

~16!

and k 5 Am 2 1 n 2 .1 in Eq. ~16!. The first term in Eq. ~13! is a sum of propagating waves; the coefficients A n and the angles, a n and b n , are unknown in general. The second term in Eq. ~13! is a sum of evanescent waves; B m , m m , and n m are unknown in general. The third term in Eq. ~13! is a continuous spectrum of plane waves; the unknown function C is continuous. See Sec. V for further comments. Let us now return to the radiation condition. Having chosen an origin O, arbitrarily, we consider a large hemisphere H r , with radius r and center O. We then require that all propagating plane-wave components v (r, u , f ; a n , b n ) in u propagate outwards through H r , away from O. This is almost built into the decomposition ~13!: we have to be careful with grazing waves @ a n 5 21 p ; see the discussion following Eq. ~8!#. A simple way to impose our radiation condition is to split the half-space z.0 and the hemisphere H r into four parts. Thus with

H

J

1 1 1 Hm r 5 ~ r, u , f ! :0< u < p , ~ m23 ! p < f , ~ m22 ! p , m51,2,3,4, 2 2 2 being the surfaces of four octants of a sphere, we require the following conditions for the regions specified: m51:

in x,0, y0, y,0,

use 2 21 p < b n ,0,

m53:

in x.0, y>0,

use 0< b n , 21 p ,

m54:

in x